# Treewidth and Minimum Fill-in of Weakly Triangulated Graphs

@inproceedings{Bouchitt1999TreewidthAM,
title={Treewidth and Minimum Fill-in of Weakly Triangulated Graphs},
author={Vincent Bouchitt{\'e} and Ioan Todinca},
booktitle={STACS},
year={1999}
}
• Published in STACS 4 March 1999
• Mathematics
We use the notion of potential maximal clique to characterize the maximal cliques appearing in minimal triangulations of a graph. We show that if these objects can be listed in polynomial time for a class of graphs, the treewidth and the minimum fill-in are polynomially tractable for these graphs. Finally we show how to compute in polynomial time the potential maximal cliques of weakly triangulated graphs.
18 Citations
Treewidth and Minimum Fill-in: Grouping the Minimal Separators
• Mathematics, Computer Science
SIAM J. Comput.
• 2001
It is shown that if these objects can be listed in polynomial time for a class of graphs, the treewidth and the minimum fill-in are polynomially tractable for these graphs.
Listing all potential maximal cliques of a graph
• Mathematics
Theor. Comput. Sci.
• 2002
Listing all potential maximal cliques of a graph
A potential maximal clique of a graph is a vertex set that induces a maximal clique in some minimal triangulation of that graph. It is known that if these objects can be listed in polynomial time for
Computing the Treewidth and the Minimum Fill-in with the Modular Decomposition
• Mathematics, Computer Science
SWAT
• 2002
Using the notion of modular decomposition, it is shown that if C is a class of graphs which is modularly decomposable into graphs that have a polynomial number of minimal separators, or graphs formed by adding a matching between two cliques, then both the treewidth and the minimum fill-in problems on C can be solved inPolynomial time.
Acyclic Colorings and Triangulations of Weakly Chordal Graphs
An acyclic coloring of a graph is a proper vertex coloring without bichromatic cycles. We show that the acyclic colorings of any weakly chordal graph G correspond to the proper colorings of
Computing the Treewidth and the Minimum Fill-In with the Modular Decomposition
• Mathematics, Computer Science
Algorithmica
• 2003
This work shows that if C is a class of graphs that are modularly decomposable into graphs that have a polynomial number of minimal separators, or graphs formed by adding a matching between two cliques, then both the treewidth and the minimum fill-in on C can be solved inPolynomial time.
Triangulated and Weakly Triangulated Graphs: Simpliciality in Vertices and Edges
• Mathematics
• 2001
We introduce the notion of weak simpliciality, in order to extend to weakly triangulated graphs properties of triangulated graphs, us ing Hayward’s notion that a vertexin a triangulated graph behaves
Recognizing Weakly Triangulated Graphs by Edge Separability
• Mathematics
Nord. J. Comput.
• 2000
A new O(m2) recognition algorithm which is not based on the notion of a 2-pair, but rather on the structural properties of the minimal separators of the graph gives the strongest relationship to the class of triangulated graphs that has been established so far.
Treewidth versus clique number. III. Tree-independence number of graphs with a forbidden structure
• Mathematics
ArXiv
• 2022
We continue the study of (tw , ω )-bounded graph classes, that is, hereditary graph classes in which the treewidth can only be large due to the presence of a large clique, with the goal of

## References

SHOWING 1-10 OF 24 REFERENCES
Treewidth of Circle Graphs
• T. Kloks
• Mathematics, Computer Science
ISAAC
• 1993
The algorithm to determine the treewidth of a circle graph can be implemented to run in O(n3) time, where n is the number of vertices of the graph.
Treewidth of Circular-Arc Graphs
• Computer Science
SIAM J. Discret. Math.
• 1994
An algorithm for computing the treewidth and constructing a corresponding tree-decomposition for circular-arc graphs in O(n^3) time is presented.
Graph Minors. II. Algorithmic Aspects of Tree-Width
• Mathematics, Computer Science
J. Algorithms
• 1986
Algorithms for Weakly Triangulated Graphs
• Mathematics, Computer Science
Discret. Appl. Math.
• 1995
Optimizing weakly triangulated graphs
• Mathematics, Computer Science
Graphs Comb.
• 1989
An algorithm which runs inO((n + e)n3) time is presented which solves the maximum clique and minimum colouring problems for weakly triangulated graphs; performing the algorithm on the complement gives a solution to the maximum stable set and minimum clique covering problems.
Algorithms for Maximum Matching and Minimum Fill-in on Chordal Bipartite Graphs
A linear time algorithm is given for the maximum matching problem and an O(n4) time algorithm for the minimum fill-in problem on chordal bipartite graphs improving previous results.
Treewidth of Chordal Bipartite Graphs
• Mathematics
J. Algorithms
• 1995
A polynomial time algorithm is presented for the exact computation of the treewidth of all chordal bipartite graphs in which every cycle of length at least six has a chord.
Algorithms for the Treewidth and Minimum Fill-in of HHD-Free Graphs
• Mathematics
WG
• 1997
Both the minimum fill-in problem and the treewidth problem are shown to be solvable in polynomial time for HHD-free graphs.
Computing Treewidth and Minimum Fill-In: All You Need are the Minimal Separators
• Mathematics, Computer Science
ESA
• 1993
It is shown that there is a polynomial time algorithm for treewidth and minimum fill-in, respectively, when restricted to the class $$\mathcal{G}$$, like permutation graphs, circle graphs, circular arc graphs, distance hereditary graphs, chordal bipartite graphs etc.